equality can mean a lot of things:
#1:White can be up a pawn but black has a positional advantage to compensate.
#2: Position is roughly equal, means that there is no counterplay for both sides.it often happen in closed positions
I don't know more examples but i hope it helped.
When playing Black, I'm not sure what the books mean by reaching equality by Black. Does it mean when White no longer has the initiative? This seems like a very vague concept to me. Would/can someone explain this in a clear fashion even I can understand?